Sequential Testing with Subadditive Costs

2025-01-29

AI Summary

This paper studies sequential fault detection under subadditive cost constraints: given that system components may fail locally and testing costs violate additivity (e.g., tree-, routing-, or machine-activation-based structures), we seek a low-cost optimal test sequence for fault identification. We propose the first approximation algorithms for three canonical subadditive cost structures. We prove that, under submodular costs, no polynomial-time logarithmic approximation exists unless the Exponential Time Hypothesis (ETH) fails. To address this, we introduce a unified optimization framework integrating truncated greedy selection, minimum-sum set cover modeling, and a dual-oracle scheme—leveraging both a ρ-approximate membership oracle and a γ-approximate value oracle—yielding a general (4ρ + γ)-approximation guarantee. Instantiating this framework yields (5 + ε)-, 9.5-, and (4 + ln n)-approximations for tree, routing, and machine-activation costs, respectively, thereby overcoming the traditional assumption of additive testing costs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

In the classic sequential testing problem, we are given a system with several components each of which fails with some independent probability. The goal is to identify whether or not some component has failed. When the test costs are additive, it is well known that a greedy algorithm finds an optimal solution. We consider a much more general setting with subadditive cost functions and provide a $(4 ho+gamma)$-approximation algorithm, assuming a $gamma$-approximate value oracle (that computes the cost of any subset) and a $ ho$-approximate ratio oracle (that finds a subset with minimum ratio of cost to failure probability). While the natural greedy algorithm has a poor approximation ratio in the subadditive case, we show that a suitable truncation achieves the above guarantee. Our analysis is based on a connection to the minimum sum set cover problem. As applications, we obtain the first approximation algorithms for sequential testing under various cost-structures: $(5+epsilon)$-approximation for tree-based costs, $9.5$-approximation for routing costs and $(4+ln n)$ for machine activation costs. We also show that sequential testing under submodular costs does not admit any poly-logarithmic approximation (assuming the exponential time hypothesis).

Problem

• Optimal Testing Sequence
• Cost-Efficient Algorithm
• Fault Identification

Innovation

• Cost-Adjusted Greedy Algorithm
• Approximate Optimal Solution
• Submodular Cost